Linear independence and orthogonality of vectors

okay, so this is really two questions, the first is kind of simple:

if a+b=0, a+c=0 and c=b can it be shown that a+b+c=0? (without the use of determinants)

The second is: what is an example of a set of vectors that are linearly independent but not orthogonal? (graphical interpretation would be appreciated)

Re: Linear independence and orthogonality of vectors

Hi guybrush92,

Knowing $\displaystyle a + b = 0$, $\displaystyle a+c=0$ and $\displaystyle b=c$ (which follows from the first two equations) it does not always follow that $\displaystyle a+b+c=0.$ Try playing around with some specific values for $\displaystyle a, b \& c$ to see why.

For the second question, thinking about $\displaystyle \mathbb{R}^{2}$ might be the easiest place to look. Geometrically, two vectors are linearly indepdent in $\displaystyle \mathbb{R}^{2}$ if they don't lie on the same line through the origin. It may be simplest to take one of the vectors to be $\displaystyle [1,0].$ To find a second vector that is linearly independent from $\displaystyle [1,0]$ and is not orthogonal to it means we should pick a vector that does not sit on the $\displaystyle x$-axis and is not perpendicular to the $\displaystyle x$-axis.

Does this get things going in the right direction? Let me know if anything is unclear. Good luck!